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march 2016 Double point-homogeneous spherical curves
Guy Valette
Bull. Belg. Math. Soc. Simon Stevin 23(1): 73-86 (march 2016). DOI: 10.36045/bbms/1457560855

Abstract

A curve is, in this paper, the image of the circle $S^1$ under an immersion $f$ into $S^2$, $\mathbb{R}^2$ or the real projective plane $P_2(\mathbb{R})$, such that every multiple point of $f$ is an ordinary double point. Such a curve $C$ is double point-homogeneous or DP-homogeneous when the group of diffeomorphisms (of $S^2, \mathbb{R}^2$ or $P_2(\mathbb{R})$) preserving $C$ has a transitive action on the set of its double points. The orbits of DP-homogeneous curves in $S^2$ are totally determined; using combinatorial methods, we prove that they fall into five countably infinite families ; the description of every family is illustrated by drawings of some representatives with a small number of double points. As a corollary, we obtain a similar classification of the DP-homogeneous curves in $\mathbb{R}^2$. We also propose a conjecture about the classification of DP-homogeneous curves in $P_2(\mathbb{R})$.

Citation

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Guy Valette. "Double point-homogeneous spherical curves." Bull. Belg. Math. Soc. Simon Stevin 23 (1) 73 - 86, march 2016. https://doi.org/10.36045/bbms/1457560855

Information

Published: march 2016
First available in Project Euclid: 9 March 2016

zbMATH: 1336.53006
MathSciNet: MR3471980
Digital Object Identifier: 10.36045/bbms/1457560855

Subjects:
Primary: 53A04
Secondary: 52C20, 58D19

Rights: Copyright © 2016 The Belgian Mathematical Society

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Vol.23 • No. 1 • march 2016
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